We define metric of the Taub-NUT-AdS black holes as [1]}, [2]}
\(ds^2 &=&-f(r)(dt+2n\cos \theta d\varphi )^2+\frac{dr^2}{f(r)}+(r^2+n^2)(d\theta ^2+\sin ^2\theta d\varphi ^2), \nonumber \\A_{\mu }&=&-[h(r)-h_0] dt+2n h(r) \cos { \theta } d\varphi ,\) 
